Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3.13·3-s + 4-s + 2.17·5-s − 3.13·6-s + 2.94·7-s + 8-s + 6.84·9-s + 2.17·10-s − 0.677·11-s − 3.13·12-s − 1.92·13-s + 2.94·14-s − 6.83·15-s + 16-s − 0.0568·17-s + 6.84·18-s − 19-s + 2.17·20-s − 9.22·21-s − 0.677·22-s − 7.41·23-s − 3.13·24-s − 0.261·25-s − 1.92·26-s − 12.0·27-s + 2.94·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.81·3-s + 0.5·4-s + 0.973·5-s − 1.28·6-s + 1.11·7-s + 0.353·8-s + 2.28·9-s + 0.688·10-s − 0.204·11-s − 0.905·12-s − 0.534·13-s + 0.786·14-s − 1.76·15-s + 0.250·16-s − 0.0137·17-s + 1.61·18-s − 0.229·19-s + 0.486·20-s − 2.01·21-s − 0.144·22-s − 1.54·23-s − 0.640·24-s − 0.0523·25-s − 0.377·26-s − 2.32·27-s + 0.555·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8018\)    =    \(2 \cdot 19 \cdot 211\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8018} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.369531354$
$L(\frac12)$  $\approx$  $2.369531354$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
19 \( 1 + T \)
211 \( 1 + T \)
good3 \( 1 + 3.13T + 3T^{2} \)
5 \( 1 - 2.17T + 5T^{2} \)
7 \( 1 - 2.94T + 7T^{2} \)
11 \( 1 + 0.677T + 11T^{2} \)
13 \( 1 + 1.92T + 13T^{2} \)
17 \( 1 + 0.0568T + 17T^{2} \)
23 \( 1 + 7.41T + 23T^{2} \)
29 \( 1 - 3.72T + 29T^{2} \)
31 \( 1 - 6.43T + 31T^{2} \)
37 \( 1 - 3.10T + 37T^{2} \)
41 \( 1 - 4.25T + 41T^{2} \)
43 \( 1 - 0.0900T + 43T^{2} \)
47 \( 1 + 9.35T + 47T^{2} \)
53 \( 1 - 9.62T + 53T^{2} \)
59 \( 1 + 2.91T + 59T^{2} \)
61 \( 1 - 7.57T + 61T^{2} \)
67 \( 1 - 7.50T + 67T^{2} \)
71 \( 1 + 4.17T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + 2.98T + 89T^{2} \)
97 \( 1 - 1.69T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.71373013696877467389656504166, −6.65214919109735274837763074426, −6.34376918947717939837451986708, −5.62819745116442886560954722488, −5.13087782445306375391718152026, −4.62608872017286268986914858394, −3.93348528730545236618366425244, −2.39822836089079339501079795827, −1.78228583780298795432815565292, −0.76903881227515801820375499648, 0.76903881227515801820375499648, 1.78228583780298795432815565292, 2.39822836089079339501079795827, 3.93348528730545236618366425244, 4.62608872017286268986914858394, 5.13087782445306375391718152026, 5.62819745116442886560954722488, 6.34376918947717939837451986708, 6.65214919109735274837763074426, 7.71373013696877467389656504166

Graph of the $Z$-function along the critical line